Big pet peeve of mine... it's LYME Disease, not Lymes Disease.
Okay, so, 20 days of antibiotics isn't going to get rid of it. It might make him sicker in the short term (Jarisch Herxeimer reaction), and in the end he'll still be sick and have long term effects. He needs AT LEAST a months worth of antibiotics. Your son should also be tested for other tick borne illnesses such as Ehrlichiosis, Babesiosis and Rocky Mountain spotted fever (if he's positive for any of those, it must be treated BEFORE starting another month of antibiotics).
Find a Lyme Literate MD (google search Lyme Literate and the place in which you live).
Watch the documentary Under Our Skin.
It is at this time that you MUST advocate for him, otherwise he could have lifelong problems. Good luck!